3.935 \(\int \frac{x^{11}}{\left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\sqrt{x^4+1}-\frac{1}{2 \sqrt{x^4+1}} \]

[Out]

-1/(2*Sqrt[1 + x^4]) - Sqrt[1 + x^4] + (1 + x^4)^(3/2)/6

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Rubi [A]  time = 0.0408458, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\sqrt{x^4+1}-\frac{1}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(1 + x^4)^(3/2),x]

[Out]

-1/(2*Sqrt[1 + x^4]) - Sqrt[1 + x^4] + (1 + x^4)^(3/2)/6

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Rubi in Sympy [A]  time = 4.04554, size = 29, normalized size = 0.76 \[ \frac{\left (x^{4} + 1\right )^{\frac{3}{2}}}{6} - \sqrt{x^{4} + 1} - \frac{1}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(x**4+1)**(3/2),x)

[Out]

(x**4 + 1)**(3/2)/6 - sqrt(x**4 + 1) - 1/(2*sqrt(x**4 + 1))

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Mathematica [A]  time = 0.0157931, size = 23, normalized size = 0.61 \[ \frac{x^8-4 x^4-8}{6 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(1 + x^4)^(3/2),x]

[Out]

(-8 - 4*x^4 + x^8)/(6*Sqrt[1 + x^4])

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Maple [A]  time = 0.006, size = 20, normalized size = 0.5 \[{\frac{{x}^{8}-4\,{x}^{4}-8}{6}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(x^4+1)^(3/2),x)

[Out]

1/6*(x^8-4*x^4-8)/(x^4+1)^(1/2)

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Maxima [A]  time = 1.43238, size = 38, normalized size = 1. \[ \frac{1}{6} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{x^{4} + 1} - \frac{1}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

1/6*(x^4 + 1)^(3/2) - sqrt(x^4 + 1) - 1/2/sqrt(x^4 + 1)

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Fricas [A]  time = 0.263039, size = 26, normalized size = 0.68 \[ \frac{x^{8} - 4 \, x^{4} - 8}{6 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

1/6*(x^8 - 4*x^4 - 8)/sqrt(x^4 + 1)

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Sympy [A]  time = 5.66389, size = 39, normalized size = 1.03 \[ \frac{x^{8}}{6 \sqrt{x^{4} + 1}} - \frac{2 x^{4}}{3 \sqrt{x^{4} + 1}} - \frac{4}{3 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(x**4+1)**(3/2),x)

[Out]

x**8/(6*sqrt(x**4 + 1)) - 2*x**4/(3*sqrt(x**4 + 1)) - 4/(3*sqrt(x**4 + 1))

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GIAC/XCAS [A]  time = 0.227877, size = 38, normalized size = 1. \[ \frac{1}{6} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{x^{4} + 1} - \frac{1}{2 \, \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

1/6*(x^4 + 1)^(3/2) - sqrt(x^4 + 1) - 1/2/sqrt(x^4 + 1)